# Thread: minkowski inequality

1. ## minkowski inequality

Let $(X,F,\mu)=([0,a],\beta, \Lambda)$ be the usual Lebesgue measurable space.
Using Holder inequality, prove Minkowski inequality for any $f \in L^2(X),g \in L^2(X)$, that is $||f+g||_2 \leq ||f||_2 +||g||_2$

i am stuck and dont know what to do. any help would be appreciated.

2. Imitate the proof of the triangle inequality. (that is start by squaring both sides) Then you can use Holder's inequality with $p=q=2$

3. Originally Posted by putnam120
Imitate the proof of the triangle inequality. (that is start by squaring both sides) Then you can use Holder's inequality with $p=q=2$
i am not sure what i should square the both sides of. holders inequality?

4. $\parallel f+g\parallel_2^2=\int (f+g)^2=\int f^2+\int g^2+2\int fg$

$(\parallel f\parallel_2 +\parallel g\parallel_2)^2=\int f^2+\int g^2+2\parallel f\parallel_2\parallel g\parallel_2$

From here you should be able to see how to use Holder's.